 
Summary: TRIANGLES OF BAUMSLAGSOLITAR GROUPS
DANIEL ALLCOCK
Abstract. Our main result is that many triangles of Baumslag
Solitar groups collapse to finite groups, generalizing a famous ex
ample of Hirsch and other examples due to several authors. A
triangle of BaumslagSolitar groups means a group with three gen
erators, cyclically ordered, with each generator conjugating some
power of the previous one to another power. There are six param
eters, occurring in pairs, and we show that the triangle fails to be
developable whenever one of the parameters divides its partner, ex
cept for a few special cases. Furthermore, under fairly general con
ditions, the group turns out to be finite and solvable of class 3.
We obtain a lot of information about finite quotients, even when
we cannot determine developability.
We study groups G of the form
(1) G(a, b; c, d; e, f) := x, y, z (xa
)y
= xb
, (yc
)z
